Schatten classes of integration operators on Dirichlet spaces

We address the question of describing the membership to Schatten-Von Neumann ideals $\mathcal{S}_ p$ of integration operators $(T_ g f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta$ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators.


Introduction and main results
Let where dA(z) = 1 π dx dy is the normalized area measure on D. For α ≥ 0, the weighted Dirichlet-type space D α consists of those functions f ∈ H(D) for which Note that the space D 0 is just the classical Dirichlet space and, as usual, will be simply denoted by D. The spaces D α are reproducing kernel Hilbert spaces: for each z ∈ D, there are functions K α z ∈ D α for which the reproducing formula f (z) = f, K α z Dα holds, where the inner product in D α is given by For 0 < p < ∞, we shall also write A p α for the weighted Bergman space of those g ∈ H(D) such that Here we put our attention on the study of the integration operator T g and the multiplication operator M g defined by where g is an analytic function on D. The bilinear operator (f, g) → f g ′ was introduced by A. Calderón in harmonic analysis in the 60's for his research on commutators of singular integral operators [8] (see also [25, p.1136]). After that, it and different variations going by the name of "paraproducts", have been extensively studied, becoming fundamental tools in harmonic analysis. Pommerenke was probably one of the first authors of the complex function theory community to consider the operator T g [17]. After the pioneering works of Aleman and Siskakis [4,5], the study of the operator T g on several spaces of analytic functions has attracted a lot of attention in recent years (see [2,3,14,16,22,23]).
Our main goal is to study the membership in the Schatten-Von Neumann ideals S p of the integration operator T g : D α → D α . If α > 1, D α is nothing else but A 2 α−2 and D 1 = H 2 , the classical Hardy space, so for p > 1, then T g ∈ S p (D α ) if and only if g belongs to the Besov space B p , and if 0 < p ≤ 1, then T g ∈ S p (D α ) if and only if g is constant (see [4,5]). We recall that, for p > 1, the Besov space B p is the space of all analytic functions g in D such that D |g ′ (z)| p (1 − |z| 2 ) p dλ(z) < ∞, where dλ(z) = dA(z) (1−|z| 2 ) 2 is the hyperbolic measure on D. The following result is implicit in the literature (see [27]) and can be proved by using the theory of Toeplitz operators (see Section 5).
Theorem A. Let g ∈ H(D). We have the following: (a) Let 0 < α < 1 and p > 1 with p(1 − α) < 2. Then T g ∈ S p (D α ) if and only if g belongs to B p . (b) If 0 < p ≤ 1 and 0 < α < 1, then T g ∈ S p (D α ) if and only if g is constant.
However for 0 < α < 1 and p(1 − α) ≥ 2, to the best of our knowledge, it is an open problem founding a description of those g ∈ H(D) such that T g ∈ S p (D α ). This motivation leads us to introduce for 0 ≤ α < ∞ and 1 < p < ∞, the space X p α which consists of those g ∈ H(D) such that (1.1) (1−|w| 2 ) p−2 dA(w) < ∞.
The following result gives a description of the membership in S p (D α ) in the range p > 1 and p(1 − α) < 4. Theorem 1. Let 0 < α < 1, g ∈ H(D) and p > 1 with p(1 − α) < 4. Then T g ∈ S p (D α ) if and only if g belongs to X p α . Now we are going to deal with the case of the classical Dirichlet space D. The situation here it seems to be more difficult. First of all, it is easy (and well known) to describe when the operator T g belongs to the Hilbert-Schmidt class S 2 (D). Indeed, for any orthonormal basis {e n } of the Dirichlet space, one has (see Section 2) (1.2) Therefore, the integration operator T g belongs to S 2 (D) if and only if the last integral in the previous equation is finite. The class of functions g ∈ H(D) satisfying this condition shall be denoted by DL.
If 1 < p < 2 Theorem A suggests that the membership in S p (D) of the operator T g could be described by those g being in the Besov space B p . However, since for p < 2 any operator on S p must be Hilbert-Schmidt, clearly the condition g ∈ DL is necessary for T g being in S p (D), and an easy calculation shows that the function g(z) = log log e 1−z belongs to B p for all p > 1 but g is not in DL. Thus, the condition g ∈ B p is not sufficient to assert that T g is in S p (D).
On the other hand, as in the weighted case, there are no trace class integration operators in the Dirichlet space unless g is constant.
Theorem 2. Let 0 < p ≤ 1 and g ∈ H(D). Then T g ∈ S p (D) if and only if g is constant.
For the case 1 < p < 2 we have a necessary condition and a different sufficient condition. We will see that they are sharp in a certain sense. Before that, for p > 1 and γ > 0, we consider the space B p,log γ , that consists of those functions g analytic on D such that . When one takes the monomials as the symbols, it turns out that the correct behavior of T g Sp is given by B p or X p 0 , while if one takes as a symbol to be functions of the type g a (z) = (1 −āz) −γ , the correct behavior is given by the B p,log p/2 condition (see Lemmas 4.1 and 4.2).
The case p > 2 seems to be a mystery. Let D p β denote the space of those functions f with f ′ ∈ A p β . For p > 2, the inclusion D p β ⊂ D holds if and only if β < (p − 2)/2; and D ⊂ D p β if and only if β ≥ p − 2 (see [28, p.94]). Thus, if one is looking for conditions on the integrability of g ′ , it can not be expected some necessary condition much better than B p = D p p−2 , and a sufficient condition must be stronger than g being in D p p−2 2 . We will discuss a little bit this case in Section 4.
We close this section saying that from now on the paper is organized as follows. In Section 2 we introduce several preliminary general results related on Schatten classes of operators on Dirichlet spaces. Section 3 is devoted to the proof of Theorem 1. There it will be proved directly (see Proposition 3.1 (iv)) the identity which together with Theorem 1 gives a proof of Theorem A not relying in the theory of Toeplitz operators. It is worth mentioning that the Besov space B p is rich of several characterizations (the identity (1.3) gives a new one), each of them being the appropriate tool to use in different situations (see [1], [7], or [29] for example). In Section 4 we prove Theorem 2 and Theorem 3. Also, by using some testing classes of functions, we show that those results are sharp in a certain sense. Finally, Section 5 is devoted to study the relationship of the integration operator T g with other classical operators acting on weighted Dirichlet spaces, such as Toeplitz operators, multiplication operators or big and small Hankel operators. A similar connection also happens in other contexts [18]. Indeed, the same techniques used in the proof of Theorem 1 work to demonstrate an extension for positive Borel measures of the helpful result of Luecking on Toeplitz operators [10, p. 347]). Throughout the paper, the letter C will denote a positive absolute constant whose value may change at different occurrences, and we write A ≍ B when the two quantities A and B are comparable.

Preliminary results
Let H and K be separable Hilbert spaces. Given 0 < p < ∞, let S p (H, K) denote the Schatten p-class of operators from H to K. If H = K we simply shall write S p (H). The class S p (H, K) consists of those compact operators T from H to K with its sequence of singular numbers λ n belonging to ℓ p , the p-summable sequence space. We recall that the singular numbers of a compact operator T are the square root of the eigenvalues of the positive operator T * T , where T * denotes the Hilbert adjoint of T . We remind the reader that T ∈ S p (H) if and only if T * T ∈ S p/2 (H). Also, the compact operator T admits a decomposition of the form T = n λ n ·, e n H σ n , where {λ n } are the singular numbers of T , {e n } is an orthonormal set in H, and {σ n } is an orthonormal set in K.
For p ≥ 1, the class S p (H, K) is a Banach space equipped with the norm , while for 0 < p < 1 one has the inequality S + T p Sp ≤ S p Sp + T p Sp . We refer to [21] or [30, Chapter 1] for a brief account on the theory of Schatten p-classes.
We shall write H for a Hilbert space of analytic functions in D with reproducing kernels K z . Given an operator T on H, usually the reproducing kernel functions carry a large amount of information about relevant properties of T , such as boundedness, compactness, membership in Schatten p-classes, etc. It is known that if {e n } is an orthonormal basis of a Hilbert space H of analytic functions in D with reproducing kernel K z , then for all z and ζ in D, see e.g. [30,Theorem 4.19]. We also introduce J z , the derivative of K z respect to z, that is, It follows that for any orthonormal set {e n } of H, and equality in (2.3) holds if {e n } is an orthonormal basis of H. We shall write k z and j z for the normalizations of these functions. In order to avoid some confusions when dealing with reproducing kernels of either D α or A 2 α , we use the notation B α z for the reproducing kernel of the weighted Bergman space A 2 α at the point z, and let b α z = B α z B α z A 2 α be its normalization. It is well known (see [30,Corollary 4.20]) that The reproducing kernel function for the Dirichlet type space D α is denoted by K α z , and k α z denotes the corresponding normalized reproducing kernel. Since f ∈ D α if and only if f ′ ∈ A 2 α , using the reproducing formula for the Bergman space A 2 α (see [30,Proposition 4.23]), it can be deduced the following expression of the reproducing kernel of D α (see [7] or [27]): In particular, for α = 0, Also, it is easy to see that The next two results are certainly well known to the experts (see [9] or [24] for similar results), but we find convenient for the reader to give a proof here.
(ii) For 0 < p ≤ 2, Proof. Since the operator T is compact, it admits the decomposition where {λ n } are the singular values of T , {e n } is an orthonormal set in A 2 α , and {f n } is an orthonormal set in H. Then If 0 < p ≤ 2, a similar argument, using Hölder's inequality with exponent 2/p ≥ 1, (2.3) and (2.4), gives The corresponding analogue of Proposition 2.1 for the Dirichlet type spaces D α uses the functions j α Proof. Since T is compact, it admits the decomposition where {λ n } are the singular values of T , {e n } is an orthonormal set in D α , and {f n } is an orthonormal set in H. It follows from (2.5) that J α z (0) = 0, then using (2.4), If p ≥ 2, using the identity (2.8), Hölder's inequality, (2.3) and (2.7) If 0 < p ≤ 2, since e n Dα = 1, and dA α (z) = (1 + α) J α z −2 Dα dλ(z) due to (2.7), then For the first term (I), observe that |λ n | ≤ T , and therefore For the second term (II), due to Hölder's inequality, (2.3) and the identity (2.8) Putting the estimates obtained for (I) and (II) in (2.9) we obtain part (ii). This completes the proof.
The following result will also be needed.
Proof. Let {e n } be any orthonormal basis of D α . From (2.2) and (2.5) we have and, since α ≥ 0, we obtain which gives the result for p = 1. If 1 < p < 2, using Hölder's inequality where the last inequality follows from (2.3) and (2.7). From here one obtains the corresponding inequality. The proof is complete.
We shall also use several times the following integral estimate (see [30]) that has become indispensable in this area of analysis.
The useful inequality which appears below is from [13], and can be thought as a generalized version of the previous one.
Lemma C. Let s > −1, r, t > 0, and r + t − s > 2. If t < s + 2 < r then, for a, z ∈ D, we have For z ∈ D and r > 0, let denote the hyperbolic disk with center z and radius r. Here β(z, w) is the Bergman or hyperbolic metric on D.
We also need the concept of an r-lattice in the Bergman metric. Let r > 0. A sequence {a k } of points in D is called an r-lattice, if the unit disk is covered by the Bergman metric disks {D k := D(a k , r)}, and β(a i , a j ) ≥ r/2 for all i and j with i = j. If {a k } is an r-lattice in D, then it also has the following property: for any R > 0 there exists a positive integer N (depending on r and R) such that every point in D belongs to at most N sets in {D(a k , R)}. There are elementary constructions of r-lattices in D. See [30,Chapter 4] for example.
Before embarking on the proof of Theorem 1, some preliminary results of interest on their own must be proved.

3.1.
A new class of spaces. In this subsection, we display several nesting properties of X p α and B p spaces. We offer a proof of (1.3), which gives under those restrictions an equivalent B p -norm. It is worth noticing that equivalent and useful B p -norms (see [1] and [7] for example) have been previously introduced for the study of operators on different spaces of analytic functions on D. Also, our next result proves that Proof. For a ∈ D fixed, let D(a) := z : |z − a| < 1−|a| 2 . (i) If g ∈ X p α , then the subharmonicity of |g ′ | 2 together with the fact that |1 −wz| ≍ (1 − |w| 2 ) for z ∈ D(w) implies that g ∈ B p . Also, since This shows that X p α ⊂ D α proving (i).
This gives and it follows easily that ||g|| q X q α ≤ C||g|| p X p α for q > p. (iii) follows from the inequality sup z∈D The inclusion X p α ⊂ B p follows from (i). Conversely, suppose that g ∈ B p . Assume first that p > 2. Since pα > p−2, we can choose ε > 0 with pα−(1+ε)(p−2) > 0. Then, using Hölder's inequality and Lemma B, we obtain Note that the choice of ε gives pα > β, and therefore we can use Lemma B again in order to obtain Now, passing the sum outside the integral and using Lemma B we get where the last step follows from Theorem 0 of [5] (see also [29]). This completes the proof.

3.2.
Proof of Theorem 1. The sufficiency for the case 1 < p ≤ 2, and the necessity for 2 ≤ p < ∞ is a byproduct of the following result, which also gives some information on the case p(1 − α) > 4.
, the result follows directly from Proposition 2.2.
The necessity for 1 < p < 2 follows from the next Proposition and part (iv) of Proposition 3.1.
Proof. Let 1 ≤ p < 2, and assume that T g ∈ S p (D α ). Then the positive operator T * g T g belongs to S p/2 (D α ). Without loss of generality we may assume that g ′ = 0. Suppose is the canonical decomposition of T * g T g . Then not only is {e n } an orthonormal set, it is also an orthonormal basis. Indeed, if there is an unit vector e ∈ D α such that e ⊥ e n for all n ≥ 1, then D |g ′ (z)| 2 |e(z)| 2 dA α (z) = T g e 2 Dα = T * g T g e, e Dα = 0 because T * g T g is a linear combination of the vectors e n . This would give g ′ ≡ 0.
Since {e n } is an orthonormal basis of D α , then by Lemma 2.3 which finishes the proof of (i). Furthermore, if T g ∈ S 1 (D α ), then (3.2) says that which implies that g is constant. This completes the proof.
The remaining part of the proof is more involved. It will be splitted in two cases.
Sufficiency. Case 2 < p ≤ 4. Let {e n } be any orthonormal set in D α . Then Since g ∈ X p α ⊂ D α by Lemma 3.1 and |e n (0)| ≤ 1, we clearly have In order to deal with the term I 2 , note first that e 2 n ∈ D 1+2α because for any f ∈ D α ,

Dα
(1 − |z|) α , z ∈ D. So from the reproducing formula for D 1+2α we deduce Therefore, if we use the notation Fubini's theorem and Hölder's inequality yields . Then, if p = 4, it follows from (2.3) and the fact that K α w 2 α . Now, if 2 < p < 4, notice that Hölder's inequality with exponent 4/p > 1 and (2.3) yield Dα . This together with the fact that for α > 0 we have K α w 2 Since g ∈ X p α combining the estimates for I 2 and I 1 we obtain that n T g e n p Dα ≤ C < ∞.
Sufficiency. Case 4 < p < ∞ and p(1 − α) < 4 . Proceeding as before we get α , and therefore we can assume that e n (0) = 0. Note that for β ≥ α we have This follows from the reproducing formula for D β and the fact that D α ⊂ D β if α ≤ β. Since pα > p − 4, we can take ε > 0 so that p .

The open case.
In relation with the open case p(1 − α) ≥ 4, we provide a result which can be proved following the lines of the proof of Theorem 1 (case p > 4), and therefore the proof will be omitted.
Obviously, X p α−ε X p α if (1 − α)p ≥ 2 (see Lemma 4.1 below), so Proposition 3.4 gives a sufficient but not necessary condition for T g ∈ S p (D α ), (1 − α)p ≥ 2. However, if α > 0 and 1 < p < ∞, those techniques which will be developed in the proof of Lemma 4.2, together with Lemma C, imply that for any β > 0, In particular, the previous result gives the right growth for this family of functions. Proof of Theorem 2. Since S p (D) ⊂ S 1 (D) for 0 < p ≤ 1, the result follows from part (ii) of Proposition 3.3.

Proof of Theorem 3. Part (a) follows from part (i) of Proposition 3.3, and part (c) is deduced in Proposition 3.2.
In order to prove part (b), assume that 1 < p < 2. Then, for all orthonormal sets {e n } of D, we have Thus, by [30, Theorem 1.27], we deduce that T g ∈ S p (D) with T g Sp ≤ C g Bp log p/2 .

4.2.
Testing functions for Schatten classes. Our next goal consists of proving that Theorem 3 gives the correct behavior of T g Sp , 1 < p < 2, at least for some families of functions. For the beginning, we deal with monomials.
Lemma 4.1. Asumme that 0 ≤ α < 1 and 1 < p < ∞. Let g j (z) = z j , j = 1, 2, 3 . . . . Then Proof. We shall use the inner product in D α given by for f (z) = ∞ n=0 a n z n , and g(z) = ∞ n=0 b n z n . We note that , n ∈ N, we have that {σ n } ∞ n=0 is an orthonomal basis of D α , and furthermore ∞ n=j a n−j n z n = ∞ n=j a n−j (n + 1) That is, the singular values of the integration operator T g j are Consequently, On the other hand, At this point, we use [12, Theorem 1] to obtain which together with (4.4) gives the first equivalence in (4.1). The second equivalence in (4.1) follows from an straightforward calculation according to those values of p and α.

Now we prove (4.2),
where in the last step we have used that ω(r) = (1 − r) p−2 log e 1−r p/2 is an admissible weight with distortion function equivalent to (1 − r) (see [15, p. 11]). Now, bearing in mind the properties of the Beta function, ≍ j (log(j + 1)) p/2 , so we get (4.2). The equivalence (4.3) can be proved analogously. This finishes the proof.
Lemma 4.2. Assume that p > 1 and γ > 0. Then , it follows that Therefore, joining this and Lemma B, On the other hand, taking 0 < ε < min(1, 2(p − 1)/p), and bearing in mind Lemma C, So, an application of Lemma B gives In order to prove (4.7), we first estimate the B p,log p/2 -norm of the functions g a (z) = (1 −āz) −γ . Take a ∈ D with |a| ≥ 1/2. Moreover, which together with (4.8) and (4.9) gives  Furthermore, if 2 ≤ p < ∞, using again (4.10) and Proposition 4.3 below, and this completes the proof of (b).
Bearing in mind that (X p α , || · || X p α ) is a Banach space for p > 1, the closed graph theorem and Lemma 4.1 and Lemma 4.2, we deduce that X p 0 B p and is different from B p,log p/2 . In particular, Proposition 3.1 (iv) does not remain true for α = 0 and 1 < p < 2.  (i) If T g ∈ S p (D) then g ∈ B p,log p/2 . (ii) If T g ∈ S p (D) then g ∈ X p 0 .

Toeplitz operators.
We recall that given a finite positive Borel measure µ on D, the Toeplitz operator Q µ on D α , α > 0 is defined by Toeplitz operators have been a key tool for studying the membership in S p of many classes of operators, such as composition operators (see [11], [10,Section 7] and [30,Chapter 11]) or integration operators (see [4,5] and [16,Chapter 6]). Indeed, the integration operator T g and the Toeplitz operator Q µ on D α are related via the identity T * g T g = Q µg , where µ g is the measure defined by dµ g (z) = |g ′ (z)| 2 dA α (z), and one can obtain a proof of Theorem A using the characterization of Schatten class Toeplitz operators obtained by D. Luecking (see (5.1) below). So, it is natural to expect that the methods used to study the membership of T g in the Schatten p-class of D α are going to work also for the Toeplitz operator Q µ on D α for a general measure µ. Before doing that, we recall Luecking's result [10] describing the membership in S p (D α ) of the Toeplitz operator Q µ for all p > 0 with p(1 − α) < 1. He shows that, for the range of p considered above, Q µ ∈ S p (D α ) if and only if, for any r-lattice {a j } with associated hyperbolic disks {D j } Given a finite positive Borel measure on D, for any −1 < α < ∞ and 0 < p < ∞ we define Here we are able to obtain a full description of the measures µ for which the Toeplitz operator Q µ belongs to S p (D α ) on the extended range of all p > 0 with p(1 − α) < 2 and 1 < p(2 + α). We remark here that, as α > 0, a complete description of the Hilbert-Schmidt Toeplitz operators on D α is obtained.
Proof. Consider the inclusion operator I µ : D α → L 2 (D, µ). It is easy to check that Q µ = I * µ I µ , and thus Q µ ∈ S p (D α ) if and only if I µ belongs to S 2p . Now, the necessity of X 2p α (µ) < ∞ for p ≥ 1 and the sufficiency for p ≤ 1 follow from Proposition 2.2. Also, by repeating the proof of the sufficiency in Theorem 1 replacing the measure |g ′ (z)| 2 dA α (z) in that proof by the measure dµ we obtain n I µ e n 2p L 2 (D,µ) ≤ C < ∞ for all orthonormal sets {e n } of D α provided p > 1 and p(1 − α) < 2. This proves the sufficiency of X 2p α (µ) < ∞ in that range. Finally, it remains to show the necessity in the case 1/(2 + α) < p < 1. Let {a j } be an r-lattice with associated hyperbolic disks {D j }. Using that |1 −wz| ≍ |1 −ā j z| for w ∈ D j and Lemma B, we deduce Thus, by Luecking's condition (5.1), if Q µ ∈ S p (D α ) then X 2p α (µ) < ∞ completing the proof of the Theorem.
We conclude this subsection mentioning that in [19] one can find a description of the membership of the Toeplitz operator Q µ in S 2k (D α ) for positive integers k in terms of some iterated integrals.

5.2.
Big and small Hankel operators. As in [26] and [20], for α ≥ 0, we consider the Sobolev space L 2 α consisting of those differentiable functions u : D → C for which the norm is finite. It is clear that D α is a closed subspace of L 2 α . Let P α be the orthogonal projection from L 2 α onto D α . The big Hankel operator H α g : D α → L 2 α and the small Hankel operator h α g : The relation between the big Hankel operator and the multiplication operator M g ′ is clear and well understood. Indeed, in [26, Corollary 1] Z. Wu shows that M g ′ : D α → A 2 α is bounded, compact, or belongs to S p with 1 < p < ∞, if and only if the same is true for the big Hankel operator H α g : D α → L 2 α . However, although Mḡ′ is related with the the small Hankel operator (see (5.4)  belongs to S p if and only if h 0 g : D → L 2 0 belongs to S p (see [26,Theorem 6]. Note that, by the previous observations, we may replace H 0 g by M g ′ or T g ). The main aim of this section consists of extending Wu's result on Schatten p-classes for the small Hankel operator to all D α and to all p with 1 < p < ∞. Before that, we recall that and has the property (see [20, p.105]) that Proof. Firstly, we recall that if T g or h α g is bounded, then g ∈ D α . It is enough to consider the relationship between Mḡ′ and h α g . For this, we look at the difference of Mḡ′ and ∂ ∂w h α g . For f ∈ D α , a straightforward calculation using that g ∈ D α and (5.3) yields For 1 < p < ∞, if T g ∈ S p (D α ) or h α g ∈ S p (D α , L 2 α ) then g ∈ B p (see Propositions 3.1, 3.2, 3.3, Theorem 3 and [26, Theorem 1]), and therefore the difference considered above, as an operator acting from D α into L 2 (D, dA α ), belongs to S p , by Proposition 5.3 (which we are going to prove below). This completes the proof.
For u ∈ L 2 (D, dA α ), consider the operator For the proof of that proposition, we need the following lemma.
Lemma 5.4. Let σ > −1, and 2 + σ < b ≤ 4 + 2σ. Then for each a ∈ D and any f ∈ H(D) we have Proof. Let ϕ a (z) = a−z 1−āz , and consider the function f a = (f • ϕ a ). After the change of variables z = ϕ a (ζ), and an application of Lemma 2.1 of [7] we Finally, the change of variables ζ = ϕ a (z) gives Proof of Proposition 5.3. Firstly we deal with the case p ≥ 2. Note that, for f ∈ H ∞ (the algebra of all bounded analytic functions on D, a dense subset of D α ) and u analytic, one has ∆ u f = uf − P α (uf ), where P α denotes the Bergman projection from L 2 (D, dA α ) to A 2 α . Therefore, ∆ u f is the solution of the equation ∂v = uf ′ with minimal L 2 (D, dA α ) norm. Now, it is well known that the solution of ∂v = uf ′ given by Indeed, the estimate in question follows from Cauchy-Schwarz inequality and the fact that, for c > 0 and t > −1, the integral D |z−w| |1−wz| 1+t+c is comparable to (1 − |z| 2 ) −c (this is just a variant of Lemma B). Taking all of this into account, we obtain that From this inequality, it follows easily that the operator ∆ u is bounded (or compact) if sup z∈D (1−|z|)|u(z)| < ∞ (or if lim |z|→1 − (1−|z|)|u(z)| = 0), and it is clear that these conditions are implied by the fact that u ∈ A p p−2 . Now, let {e n } be any orthonormal set in D α . Therefore, using (5.5), Hölder's inequality, (2.3) and (2.7), we obtain A different proof for the case p = 2 (that can be adapted to the case p > 2) can be given as follows. Let {e n } be any orthonormal basis of D α . Take 0 < ε < 1. Then, Lemma 5.4 yields Therefore, using (2.3) and Lemma B, we get n ∆ u e n 2 = n D |∆ u e n (w)| 2 dA α (w) For 1 < p < 2, one has A p p−2 ⊂ A 2 . Thus, by the case we have just proved, the operator ∆ u is Hilbert-Schmidt and, in particular, compact. By Proposition 2.2, a sufficient condition for ∆ u to be in the class S p is Now, take 0 < ε < 1 with α − ε > −1 and p − εp > 1. Proceeding as in (5.6), and then using Lemma C we obtain This, together with Lemma C, gives Thus, (1 − |z| 2 ) p−2−εp dA(z).

Multiplication operators.
It is well known that the multiplication operator M g ′ : D α → A 2 α is bounded or compact if and only if M g ′′ : D α → A 2 2+α is bounded or compact. Thus, a natural question arises here: It is true that M g ′ : D α → A 2 α is in the Schatten class S p if and only if M g ′′ : D α → A 2 2+α belongs to S p ? We are going to see that this happens when p > 1, but the result is false for p = 1. Let us consider the spaceṡ A 2 α = {f ∈ A 2 α : f (0) = 0} andḊ α = {f ∈ D α : f (0) = 0}.
Theorem 5.5. Let α ≥ 0, 1 < p < ∞ and g ∈ H(D). The following are equivalent: (a) M g ′ :Ḋ α →Ȧ 2 α is in S p ; (b) M g ′′ :Ḋ α → A 2 2+α is in S p . Taking into account Theorems A and 2, the next result shows that it is no longer true that M g ′ being in the trace class S 1 is equivalent to M g ′′ being in the trace class. We recall that g ∈ B 1 if g ∈ H(D) and D |g ′′ (z)| dA(z) < ∞. Moreover, there is a function g ∈ H(D) with M g ′′ ∈ S 1 (D, A 2 2 ) such that D |g ′′ (z)| ϕ(z) dA(z) = ∞ for any function ϕ(r) increasing continuously to ∞ on (0, 1).
One should compare Theorem 5.6 with the results obtained in Theorem 8 of [6], where trace class bilinear Hankel forms on the Dirichlet space are studied.
Proof of Theorem 5.5. We recall that if (a) or (b) holds, then g ∈ B p . We first deal with the case p ≥ 2. Since f A 2 α ≍ f ′ and we deduce , and by what we have just proved (see (5.11) and the comments after that), one gets M g ′′ a S 1 (D,A 2 2 ) ≤ C(1 − |a| 2 ) −γ log e 1 − |a| 2 and putting this into (5.13) gives This shows together with part (b) that given a lacunary series g(z) = k a k z n k , the multiplication operator M g ′′ : D → A 2 2 belongs to S 1 if and only if k n k |a k | < ∞, and it is well known that this condition is equivalent to g being in B 1 [30, p. 100]. Now, given a function ϕ as described in part (d), it is straightforward to select the numbers {a k } and the sequence {n k } so that the summability condition k n k |a k | < ∞ is met, but D |g ′′ | ϕ dA = ∞.